We describe experimental investigations of the structure of two-dimensional spherical crystals. The crystals, formed by beads self-assembled on water droplets in oil, serve as model systems for exploring very general theories about the minimum energy configurations of particles with arbitrary repulsive interactions on curved surfaces. Above a critical system size we find that crystals develop distinctive high-angle grain boundaries, or scars, not found in planar crystals. The number of excess defects in a scar is shown to grow linearly with the dimensionless system size. The observed slope is expected to be universal, independent of the microscopic potential

Bausch, A. R.

Bowick, Mark

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Syracuse University SURFACE Physics College of Arts and Sciences 3-15-2003 Grain Boundary Scars and Spherical Crystallography Mark Bowick Department of Physics, Syracuse University, Syracuse, NY A. R. Bausch Technische Universitat Munchen Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Bowick, Mark and Bausch, A. R., "Grain Boundary Scars and Spherical Crystallography" (2003). Physics. 155. https://surface.syr.edu/phy/155 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:cond-mat/0303289v1  [cond-mat.soft]  15 Mar 2003Grain Boundary Scars and Spherical CrystallographyA. R. Bausch∗, M. J. Bowick†, A. Cacciuto‡, A. D. Dinsmore§, M. F.Hsu¶, D. R. Nelson¶, M. G. Nikolaides∗¶, A. Travesset‖ & D. A. Weitz¶∗ Department of Physics, E22, Technische Universität München, 85747 München, Germany† Physics Department, Syracuse University, Syracuse NY 13244-1130, USA‡ FOM Institute for Atomic and Molecular Physics,Kruislaan 407, 1098 SJ, Amsterdam, The Netherlands§ Department of Physics, University of Massachusetts, Amherst, MA 01003-4525, USA¶ Department of Physics and DEAS, Harvard University, Cambridge MA 02138, USA and‖ Physics and Astronomy Department, Iowa State and Ames Natl. Lab, Ames, IA 50011, USAWe describe experimental investigations of the structure of two-dimensional spher-ical crystals. The crystals, formed by beads self-assembled on water droplets in oil,serve as model systems for exploring very general theories about the minimum energyconfigurations of particles with arbitrary repulsive interactions on curved surfaces.Above a critical system size we find that crystals develop distinctive high-angle grainboundaries, or scars, not found in planar crystals. The number of excess defects in ascar is shown to grow linearly with the dimensionless system size. The observed slopeis expected to be universal, independent of the microscopic potential.Spherical particles on a flat surface pack most effi-ciently in a simple lattice of triangles, similar to billiardballs at the start of a game. Such six-fold coordinatedtriangular lattices [1] cannot, however, be wrapped onthe curved surface of a sphere; instead, there must beextra defects in coordination number. Soccer balls andC60 fullerenes [2, 3] provide familiar realizations of thisfact – they have 12 pentagonal panels and 20 hexago-nal panels. The necessary packing defects can be char-acterized by their topological or disclination charge, q,which is the departure of their coordination number cfrom the preferred flat space value of 6 (q = 6 − c); aclassic theorem of Euler [4, 5] shows that the total discli-nation charge of any triangulation of the sphere must be12 [6]. A total disclination charge of 12 can be achievedin many ways, however, which makes the determinationof the minimum energy configuration of repulsive par-ticles, essential for crystallography on a sphere, an ex-tremely difficult problem. This was recognized nearly100 years ago by J.J. Thomson [7], who attempted, un-successfully, to explain the periodic table in terms of rigidelectron shells. Similar problems recur in fields as diverseas multi-electron bubbles in superfluid helium [8], virusmorphology [9, 10, 11], protein s-layers [12, 13] and cod-ing theory [14, 15]. Indeed, both the classic Thomsonproblem, which deals with particles interacting throughthe Coulomb potential, and its generalization to other in-teraction potentials remain largely unsolved after almost100 years [16, 17, 18].The spatial curvature encountered in curved geome-tries adds a fundamentally new ingredient to crystallog-raphy, not found in the study of order in spatially flat sys-tems. To date, however, studies of the Thomson and re-lated problems have been limited to theory and computersimulation. As the number of particles on the spheregrows, isolated charge 1 defects are predicted to inducetoo much strain; this can be relieved by introducing ad-ditional dislocations, consisting of pairs of tightly bound5-7 defects [19] which still satisfy Euler’s theorem sincetheir net disclination charge is zero. Dislocations, whichare point-like topological defects in two dimensions, dis-rupt the translational order of the crystalline phase butare less disruptive of orientational order [19]. While theyplay an essential role in crystallography on a sphericalsurface, the configuration and orientation of these ex-cess defects remains undetermined, and can only be fullyunderstood through a combination of theory and exper-iment. Experimental realizations that probe the subtlestructures have, however, been sorely lacking.We present an experimental realization of the gener-alized Thomson problem that allows us to explore thelowest energy configuration of the dense packing of re-pulsive particles on a spherical surface. We create twodimensional packings of colloidal particles on the sur-face of spherical water droplets, and view the structureswith optical microscopy. Above a critical system size, thethermally equilibrated colloidal crystals display distinc-tive high-angle grain boundaries, which we label “scars”.These grain boundaries are found to end within the crys-tal, which is not observed to occur on flat surfaces be-cause the energy penalty is too high.Our experimental system is based on the self-assemblyof one micron diameter cross-linked polystyrene beadsadsorbed on the surface of spherical water droplets (ofradius R), themselves suspended in a density-matchedtoluene/chlorobenzene mixture [20]. The particles areimaged with phase contrast using an inverted microscope.The curvature of the spherical water droplet limits theimaged surface area to between 5% and 20% of the fullsurface area of the sphere, depending on the size of thedroplet. After determining the center of mass of eachbead, the lattice geometry is analyzed by Delaunay tri-2angulation algorithms [21] appropriate to spherical sur-faces.We analyze the lattice configurations of a collectionof 40 droplets. A typical small spherical droplet withsystem size, R/a = 4.2, where a is the mean particlespacing, is shown in Fig. 1A. The associated Delaunaytriangulation is shown in Fig. 1B. The only defect is oneisolated charge +1 disclination. Extrapolation to the en-tire surface of the sphere is statistically consistent withthe required 12 total disclinations.Qualitatively different results are observed for largerdroplet sizes as defect configurations with excess dislo-cations appear. Although some of these excess dislo-cations are isolated, most occur in the form of distinc-tive (5 − 7 − 5 − 7 − · · · − 5) chains, each of net charge+1, as shown in Fig. 1D. These chains form high-angle(30◦) grain boundaries, or scars, which terminate freelywithin the crystal. Such a feature is energetically pro-hibitive in equilibrium crystals in flat space. Thus, al-though grain boundaries are a common feature of 2Dand 3D crystalline materials, arising from a mismatchof crystallographic orientations across a boundary, theyusually terminate at the boundary of the sample in flatspace because of the excessive strain energy associatedwith isolated terminal disclinations. Termination withinthe crystal is a feature unique to curved space. Thus,our results provide important guidance to determine theconfiguration of excess defects on a sphere.To ascertain that the colloidal particles are equili-brated, we use particle tracking routines and subsequentautomated triangulation to measure the mobility of theobserved screening dislocations. The diffusion of ther-mally excited colloid particles on the surface of the wa-ter droplets results in local rearrangements of the crystalstructure. Thermal fluctuations create and destroy dis-locations once every few seconds, on average, indicatingthat the defect arrays reach equilibrium much faster thanthe observation time of 10-60 min. The equilibration timecan also be estimated by the time required for a dislo-cation to diffuse across typical defect structures. Themeasured diffusion constant allows us to calculate equili-bration times that range from a few seconds to hundredsof seconds, depending on the size of the crystal. Since thespherical crystals exist for 10 to 60 minutes, the systemhas sufficient time to reach equilibrium. Thus, our obser-vations reflect the equilibrium ground state, as opposedto a history-dependent nonequilibrium effect, which isthe case for crystals in flat space.To quantify the behavior of the scars, we determine thenumber of excess dislocations per chain for each droplet,with the convention that dislocations are counted as be-ing part of the same array if they are within three lat-tice spacings, and plot the results as a function of R/ain Fig. 2. Scars only appear for droplets with R/a ≥ 5.These results provide a critical confirmation of a theoret-ical prediction that R/a must exceed a threshold valueFIG. 1: Light microscope images of particle-coated droplets.Two droplets (A) and (C) are shown, together with theirassociated defect structures (B) and (D). Panel (A) showsan ≈ 13% portion of a small spherical droplet with radiusR = 12.0 microns and mean particle spacing a = 2.9 mi-crons (R/a = 4.2), along with the associated triangulation(B). Charge +1(−1) disclinations are shown in red and yel-low respectively. Only one +1 disclination is seen. Panel (C)shows a cap of spherical colloidal crystal on a water droplet ofradius R = 43.9 microns with mean particle spacing a = 3.1microns (R/a = 14.3), along with the associated triangulation(D). In this case the imaged crystal covers about 17% of thesurface area of the sphere. The scale bars in (A) and (C) are5 microns.FIG. 2: Excess dislocations as a function of system size. Thenumber of excess dislocations per minimal disclination N as afunction of system size R/a, with the linear prediction givenby theory shown as a solid red line.(R/a)c ≈ 5, corresponding to M ≈ 360 particles, for ex-cess defects to proliferate in the ground state of a spher-ical crystal [22]. The precise value of (R/a)c depends ondetails of the microscopic potential but its origin is easilyunderstood by considering just one of the 12 charge +1disclinations required by the topology of the sphere. Inflat space such a topological defect has an associated en-ergy that grows quadratically with the size of the system,3since it is created by excising a 60◦ wedge of material andgluing the boundaries together [23]. The elastic strain en-ergy associated with this defect grows as the area. In thecase of the sphere the radius plays the role of the systemsize. As the radius increases, isolated disclinations be-come much more energetically costly. This elastic strainenergy may be reduced by the formation of linear dislo-cation arrays, i.e. grain boundaries. The energy neededto create these additional dislocation arrays is propor-tional to a dislocation core energy Ec and scales linearlywith the system size. Such screening is inevitable in flatspace (the plane) if one forces an extra disclination intothe defect-free ground state. Unlike the situation in flatspace, grain boundaries on the sphere can freely termi-nate [22, 24, 25, 26, 27], and our experimental resultsconfirm these theoretical expectations.One systematic approach to determining the groundstate of a collection of M particles distributed on thesphere and interacting via an arbitrary repulsive poten-tial [22, 27] treats the disclination defects themselvesas the fundamental degrees of freedom, with the 6-coordinated particles forming a continuum elastic back-ground. The agreement between the predicted and ob-served values of (R/a)c supports the validity of this the-oretical approach. The original particle pair potentialis replaced by a long range defect pair potential givenby χ(β) ∝ R2(1 +∫1−cosβ20dz lnz1−z ), for a pair of defectsseparated by an angular distance β. The potential isattractive for opposite charged defects and repulsive forlike-charged defects. The underlying microscopic poten-tial enters only in determining the proportionality con-stant (equivalent to an elastic Young modulus) and Ec.Many predictions of this model are therefore universalin the sense that they are insensitive to the exact micro-scopic potential. This enables us to make definite predic-tions even though the colloidal potential is not preciselyknown. It also means that our model system serves asa prototype for any analogous system with repulsive in-teractions and spherical geometry. To further test thevalidity of this approach, we show a typical ground statefor large M in Fig. 3. The system size here is R/a = 12,similar to the droplet in Fig. 1D. The results are remark-ably similar to the experimentally observed configurationin Fig. 1D; the only difference is a result of thermal fluc-tuations, which break the two defect scars in the exper-iment. This agreement between theory and experimentalso provides convincing evidence that these scars are es-sential components of the equilibrium crystal structureon a sphere.The theory predicts that an isolated charge +1 discli-nation on a sphere is screened by a string of dislocationsof length cos−1(5/6)R ≈ 0.59R [22]. We can use the vari-able linear density of dislocations to compute the totalnumber of excess dislocations N in a scar. We find thatN grows for large (R/a) as π3[√11 − 5 cos−1(5/6)]Ra≈FIG. 3: Model grain boundaries. This image is obtained froma numerical minimization, based on the theory of [22], for asystem size comparable to the droplet in Figs. 1(C,D).0.41(Ra), independently of the microscopic potential.This prediction is universal, and is in remarkable agree-ment with the experiment, as shown by the solid line inFig. 2.We expect these scars to be widespread in nature.They should occur, and hence may be exploited, insufficiently large viral protein capsids, giant spher-ical fullerenes, spherical bacterial surface layers (s-layers) and the siliceous skeletons of spherical radiolaria(aulosphaera)[28], provided that the spherical geometryis not too distorted. Terminating strings of heptagonsand pentagons might serve as sites for chemical reactionsor even initiation points for bacterial cell division [12] andwill surely influence the mechanical properties of spheri-cal crystalline shells.[1] For the case of a Coulomb potential, see L. Bonsall, A.A.Maradudin, Phys. Rev. B 15, 1959 (1977).[2] H.W. Kroto et al., Nature 318, 162 (1985).[3] M.F. Jarrold, Nature 407, 26 (2000).[4] L. Euler, Opera Omnia, series i, Vol.26, Orell Füssli Verlag, 1953 (see alsohttp://www.ics.uci.edu/∼eppstein/junkyard/euler/).[5] P. Hilton, J. Pedersen, Amer. Math. Monthly 103, 121(1996).[6] 12 here is obtained from 6χ, where the Euler character-istic χ of a space is a topological invariant equal to 2 forthe sphere.[7] J.J. Thomson, Phil. Mag. 7, 237 (1904).[8] P. Leiderer, Z. Phys. B 98, 303 (1995).[9] D.L.D. Caspar, A. Klug, Cold Spring Harbor Symposiaon Quantitative Biology Vol. XXVII (Basic Mechanismsin Animal Virus Biology), 1 (1962).[10] C.J. Marzec, L.A. Day, Biophys. Jour. 65, 2559 (1993).[11] V.J. Reddy et al., J. Virol. 75, 11943 (2001) (see alsohttp://mmtsb.scripps.edu/viper/viper.html).4[12] U.B. Sleytr, M. Sára, D. Pum, B. Schuster, Prog. Surf.Sci. 68, 231 (2001).[13] D. Pum, P. Messner, U.B. Sleytr, J. Bacteriology 173,6865 (1991).[14] N.J.A. Sloane, Sci. Am. 250, 116 (1984).[15] J.H. Conway, N.J.A. Sloane, Sphere Packings, Latticesand Groups (Third Edition, Springer-Verlag, New York,1998).[16] The determination of the asymptotics of the ground stateenergy of spherical crystals with a logarithmic potentialappeared as Problem 7 of Smale’s list of key mathemat-ical problems for the 21st century: S. Smale, Math. In-telligencer 20, No. 2, 7 (1998).[17] E.L. Altschuler et. al., Phys. Rev. Lett. 78, 2681 (1997).[18] T. Erber, G.M. Hockney, Advances in Chemical Physics(eds I. Prigogine, S.A. Rice) Vol. XCVIII, 495 (1997).[19] D.R. Nelson, Defects and Geometry in Condensed Mat-ter Physics, (Cambridge University Press, Cambridge,2002).[20] A.D. Dinsmore et. al., Science 298, 1006 (2002); theparticular system we explored is obtained by confiningcross-linked polystyrene beads (IDC, Portland, USA) tothe surface of water droplets.[21] B. Delaunay, Bull. Acad. Sci. USSR: Class. Sci. Math.Nat. 7, 793 (1934).[22] M.J. Bowick, D.R. Nelson, A. Travesset, Phys. Rev. B62, 8738 (2000).[23] P.M. Chaikin, T.C. Lubensky, Principles of CondensedMatter Physics, (Cambridge University Press, Cam-bridge, 1995).[24] M.J.W. Dodgson, M.A. Moore, Phys. Rev. B 55, 3816(1997).[25] A. Perez-Garrido, M.A. Moore, Phys. Rev. B 60, 15628(1999) and references therein.[26] A. Toomre, (private communication).[27] M. Bowick, A. Cacciuto, D.R. Nelson, A. Travesset,Phys. Rev. Lett. 89, 185502 (2002).[28] O.R. Anderson, Radiolaria (Springer-Verlag, New York,1983).AcknowledgementsThis work was supported by the Department of En-ergy, the National Science Foundation, the Harvard Ma-terials Research Science and Engineering Center (DMR-0213805), the Fonds der Chemischen Industrie and theEmmy Noether Programme of the DFG.Correspondence and requests for materials should be ad-dressed to MJB (email: bowick@physics.syr.edu) or ARB(email:abausch@ph.tum.de).

Grain Boundary Scars and Spherical Crystallography

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Grain Boundary Scars and Spherical Crystallography

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